The discussion revolves around incorporating a step-wise potential into the Schrödinger equation for a one-dimensional box. The potential is defined as V(x) = -Vo exp(-x/L) for 0 < x ≤ L and V(x) = ∞ for x ≤ 0. Participants are exploring how to set up the equation given these boundary conditions.
Some participants have suggested solving the Schrödinger equation in each region and matching the solutions at the boundaries. There is ongoing clarification about the implications of the potential outside the specified range and the necessity of boundary conditions.
Participants note that the conditions for x < 0 and x > L are not explicitly defined, leading to questions about their relevance in the context of the problem. There is also mention of needing to use the TISE to derive a general form of the wave function and solve for constants based on boundary conditions.
What about ##x> L##?Litmus said:* Boundary Conditions: (potential)
V(x)=-Vo exp(-x/L) for 0<x≤L
V(x)=∞ for x≤0
What about x>L?
And what can you say about the wave function for x<0?
vela said:You need to solve the Schrödinger equation in each region and then match the solutions at the boundaries. Your book should have examples of how to do this.
Litmus said:I don't understand. I've given conditions x<0, and x>L it's not specified. Why do we care?
Like this?
(-h^2 / 2m ) (d^2 ψ / dx^2) + [-Vo exp(-x/L) ] ψ = E * psi
(-h^2 / 2m ) (d^2 ψ / dx^2) + ∞ψ = E * psi
Boundary is 0 so:
(-h^2 / 2m ) (d^2 ψ / dx^2) + [-Vo] ψ = E * psi
... how do I proceed?